**Math 5334: Differential Geometry**

**Course Description**: This course covers both the
differential and the geometric parts of differential geometry. The differential part includes manifolds, the
tangent bundle of a manifold, differential forms, tensors, and Riemannian
metrics. All of this is used in
formulating the geometric part, which includes curvature and torsion of space
curves, Gaussian curvature of surfaces, the Gauss-Bonnet Theorem, covariant
derivatives, parallel transport, and the Riemann curvature tensor on manifolds
of higher dimension.

**List of Topics**:

1. Manifolds

Examples of manifolds

Coordinate charts

Transition functions between coordinate
charts

Atlases on manifolds

Differentiability of functions between
manifolds

The Jacobian

The chain rule

2. The
Tangent Bundle

Description of vectors as linear operators

Transformations of vectors under changes of
coordinates

Vector fields

3. The
Cotangent Bundle

Differential 1-forms

Transformations of differential forms under a change of coordinates

4. Tensors

Covariant and contravariant
tensors

Raising and lowering of indices (via a
Riemannian metric)

5. Inner
Products

The matrix of an inner product with respect
to a basis

Riemannian metrics on manifolds

6. Curves in
the Plane and in Space

Curvature and torsion of plane curves and
space curves

The Serret-Frenet
formulas

7. Curvature
of Surfaces in Space

Gaussian curvature

The first and second fundamental forms

The Theorema Egregium

8.
Differential Geometry on Higher-dimensional Manifolds

Riemann curvature tensor (its definition
and meaning)

Torsion tensor (its definition and meaning)

Covariant derivatives and the Levi-Civita connection

Parallel transport

9. The
Gauss-Bonnet Theorem

**References:**

1. Michael Spivak, **A**** Comprehensive
Introduction to Differential Geometry**, Volumes I and II (and Vol. III for
the Gauss-Bonnet Theorem)

2. Manfredo Do Carmo**, Differential Geometry of Curves and
Surfaces**

3.** **Michael Spivak,** Calculus on Manifolds**

4. M.
Nakahara,** Geometry and Physics**